Best Known (55−17, 55, s)-Nets in Base 5
(55−17, 55, 252)-Net over F5 — Constructive and digital
Digital (38, 55, 252)-net over F5, using
- 1 times m-reduction [i] based on digital (38, 56, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 28, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 28, 126)-net over F25, using
(55−17, 55, 518)-Net over F5 — Digital
Digital (38, 55, 518)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(555, 518, F5, 17) (dual of [518, 463, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(555, 632, F5, 17) (dual of [632, 577, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- linear OA(553, 625, F5, 17) (dual of [625, 572, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(549, 625, F5, 16) (dual of [625, 576, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(545, 625, F5, 14) (dual of [625, 580, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(50, 5, F5, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(51, 2, F5, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(16) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(555, 632, F5, 17) (dual of [632, 577, 18]-code), using
(55−17, 55, 49161)-Net in Base 5 — Upper bound on s
There is no (38, 55, 49162)-net in base 5, because
- 1 times m-reduction [i] would yield (38, 54, 49162)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 55 511684 027269 261424 941279 354840 968065 > 554 [i]